Mathematical notation

Etymology of some common mathematical terms

The notation and terminology for powers and exponents is interesting. Power is first used for the square. Euclid uses the phrase in power , for example he says that magnitudes are commensurable in power when their squares are commensurable. Of course Euclid thought geometrically and the square to him was the geometrical square not a new number formed by multiplying the number by itself. Henry Billingsley, the first English translator of Euclid in 1570, makes the definition precise in his translation of Euclid's Second book:-
The power of a line is the square of the same line.
Similarly, Thomas Digges, another English mathematician, in his book Pantom of 1571 gives this definition (note the rather interesting old English):
A lyne is sayde to be equall in power with two or more lynes when his square is equall to all their squares.
Jeake, in his Arithmetic of 1696 (written in 1674), uses the word indices for the first time.
Mark their indices or how many degrees the Number you would produce is removed from the Root as whether it be second, third, fourth, etc.
In the same book Jeake also uses powers for exponents greater than 2.
Multiply alternately .... the numbers given by the powers of these alternate indices for the reduced surds.
As to the notation, Chuquet in Triparty wrote 5, 5^1, 5^2, 5^3 where we would write 5, 5x, 5x^2, 5x^3. Although written in the latter part of the 15 C it was not published until 1880.

Heinrich Schreyber (1521) wrote:-

When now such a number is to be written after another according to a proportion, then write each such quantity with the number of its order, so that in the case of double proportion the number 1 is placed over 2, 2 over 4,.....
             1  2  3   4   5   6    7  ...     16
             2  4  8  16  32  64  128  ...  65536
In the same book Schreyber gives a table for dividing powers. In this table he writes 2a for a^2, 3a for a^3 etc. He gives 8a divided by 5a is 3a etc. and 5a divided by 8a as 5a/8a (in the actual book this is 5a with a line below it and 8a below the line. Hence Schreyber wrote his powers as operators on the left - perhaps he was an analyst rather than an algebraist !

Michael Stifel in Arithmetica integra (1544) extended the powers of 2 given above by Schreyber to the left so that he had negative exponents, so -1 is written above 1/2, -2 above 1/4 etc.

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JOC/EFR May 2017

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School of Mathematics and Statistics
University of St Andrews, Scotland